slides bioinformatique1112 chapter4 reduced · bioinformatics chapter 4: sequence comparison

Bioinformatics Chapter 4: Sequence comparison

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B I O I N F O R M A T I C S

Kristel Van Steen, PhD2

Montefiore Institute - Systems and Modeling

GIGA - Bioinformatics

ULg

[email protected]


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CHAPTER 4: SEQUENCE COMPARISON

1 The biological problem

Paralogs and homologs

2 Pairwise alignment

3 Global alignment

4 Local alignment


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5 Rapid alignment methods

5.a Introduction

5.b Search space reduction

5.c Binary searches

5.d FASTA

5.e BLAST


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1 The biological problem

Introduction

• Ever-growing data sequence

data bases make available a

wealth of data to explore.

- Recall that these data bases

have a tendency of doubling

approximately every 14

months and that

- they comprise a total of over

11 billion bases from more

than 100,000 species


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Introduction

• Much of biology is based on recognition of shared characters among

organisms

• The advent of protein and nucleic acid sequencing in molecular biology

made possible comparison of organisms in terms of their DNA or the

proteins that DNA encodes.


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Introduction

• These comparisons are important for a number of reasons.

- First, they can be used to establish evolutionary relationships

among organisms using methods analogous to those employed for

anatomical characters.

- Second, comparison may allow identification of functionally

conserved sequences (e.g., DNA sequences controlling gene

expression).

- Finally, such comparisons between humans and other species may

identify corresponding genes in model organisms, which can be

genetically manipulated to develop models for human diseases.


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Evolutionary basis of alignment

• This lies in the fact that sequence alignment enables the researcher to

determine if two sequences display sufficient similarity to justify the

inference of homology.

• Understanding the difference between similarity and homology is of utmost

importance:

- Similarity is an observable quantity that may be expressed as a % identity

or some other measure.

- Homology is a conclusion drawn from the data that the two genes share a

common evolutionary history. (S-star Subbiah)


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Evolutionary basis of alignment

• Genes are either homologous or not homologous:

- they either share or

- do not share a common evolutionary history.

• There are no degrees of homology as are there in similarity.

• While it is presumed that the homologous sequences have diverged from a

common ancestral sequence through iterative molecular changes we do

not actually know what the ancestral sequence was.

• In contrast to homology, there is another concept called homoplasy:

Similar characters that result from independent processes (i.e.,

convergent evolution) are instances of homoplasy (S-star Subbiah)


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Homology versus similarity

• Hence, sequence similarity is not sequence homology and can occur by

chance ...

• If two sequences have accumulated enough mutations over time, then the

similarity between them is likely to be low.

• Consequently, homology is more difficult to detect over greater

evolutionary distances


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Orthologs and paralogs

• We distinguish between two types of homology

- Orthologs: homologs from two different species, separated by a, what

is called, a speciation event

- Paralogs: homologs within a species, separated by a gene duplication

event.


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Orthologs and paralogs

• Orthologs typically retain the original function

• In paralogs, one copy is free to mutate and acquire new function.



Example of paralogy: hemoglobin

• Hemoglobin is a protein complex

that transports oxygen

• In adults, 3 types are normally

present:

- Hemoglobin A

- Hemoglobin A2

- Hemoglobin F

each with a different

combination of subunits.

• Each subunit is encoded by a

separate gene.



Example of paralogy: hemoglobin

• The subunit genes are paralogs of

each other. In other words, they

have a common ancestor gene

• Check out the hemoglobin human

paralogs using the NCBI sequence

databases:

http://www.ncbi.nlm.nih.gov/sites/entrez

?db=nucleotide



Sickle cell disease

� Sickle-cell disease, or sickle-cell

anaemia (or drepanocytosis), is a

life-long blood disorder

characterized by red blood cells

that assume an abnormal, rigid,

sickle shape.

� Sickling decreases the cells'

flexibility and results in a risk of

various complications.

(Wikipedia)

� The sickling occurs because of a

mutation in the hemoglobin gene.

Life expectancy is shortened, with

studies reporting an average life

expectancy of 42 and 48 years for

males and females, respectively



Example of orthology: insulin

• The genes that code for insulin in humans (Homo Sapiens) and mouse (Mus

musculus) are orthologs

• They have a common ancestor gene in the ancestor species of human

and mouse

(example of a phylogenetic tree - Lakshmi)



Structural basis for alignment (as opposed to evolutionary basis)

• It is well-known that when two protein sequences have more than 20-30%

identical residues aligned the corresponding 3-D structures are almost

always structurally very similar.

• Therefore, sequence alignment is often an approximate predictor of the

underlying 3-D structural alignment

• Form often follows function. So sequence similarity by way of structural

similarity implies similar function.

(S-star Subbiah)



2 Pairwise alignment

Introduction

• An important activity in biology is identifying DNA or protein

sequences that are similar to a sequence of experimental interest,

with the goal of finding sequence homologs among a list of similar

sequences.

• By writing the sequence of gene gA and of each candidate homolog as

strings of characters, with one string above the other, we can

determine at which positions the strings do or do not match.

• This is called an alignment.

Aligning polypeptide sequences with each other raises a number of additional issues

compared to aligning nucleic acid sequences, because of particular constraints on

protein structures and the genetic code (not covered in this class).



Introduction

• There are many different ways that two strings might be aligned.

Ordinarily, we expect homologs to have more matches than two

randomly chosen sequences.

• The seemingly simple alignment operation is not as simple as it

sounds.

• Example (matches are indicated by . and - is placed opposite bases

not aligned):



Introduction

• We might instead have written the sequences

• We might also have written

Which alignment is better?

What does better mean?



Introduction

• Next, consider aligning the sequence TCTAG with a long DNA sequence:

• We might suspect that if we compared any string of modest length

with another very long string, we could obtain perfect agreement if

we were allowed the option of "not aligning" with a sufficient number

of letters in the long string.

• Don’t we prefer some type of parsimonious alignment? • What is parsimonious?



Introduction

• The approach adopted is guided by biology

NIH – National Human Genome

Research Institute)



Introduction

• Mutations such as segmental duplication, inversion, translocation often

involve DNA segments larger than the coding regions of genes.

• Point mutations, insertion or deletion of short segments are important in

aligning targets whose size are less than or equal to the size of coding

regions of genes

• Therefore, these insertions and deletions need to be explicitly

acknowledged in the alignment process.



Introduction

• There are multiple ways of aligning two sequence strings, and we may

wish to compare our target string (target meaning the given sequence

of interest) to entries in databases containing more than 107

sequence entries or to collections of sequences that are billions of

letters long.

• How do we do this?

- We differentiate between alignment of two sequences with each

other

� which can be done using the entire strings (global alignment )

� or by looking for shorter regions of similarity contained within

the strings (local alignment)

- and multiple-sequence alignment (alignment of more than two

strings)



Introduction

• Consider again the following alignment; we would like to acknowledge

insertions and deletions when “aligning”

• We can't tell whether the string at the top resulted from the insertion

of G in ancestral sequence ACTCTAG or whether the sequence at the

bottom resulted from the deletion of G from ancestral sequence

ACGTCTAG.

• For this reason, alignment of a letter opposite nothing is simply

described as an indel (i.e., insertion/deletion).



Toy example, using the sequence as a whole

• Suppose that we wish to align WHAT with WHY.

• Our goal is to find the highest-scoring alignment. This means that we

will have to devise a scoring system to characterize each possible

alignment.

• One possible alignment solution is

WHAT

WH-Y

• However, we need a rule to tell us how to calculate an alignment

score that will, in turn, allow us to identify which alignment is best.

• Let's use the following scores for each instance of match, mismatch,

or indel:

- identity (match) +1

- substitution (mismatch) -μ

- indel - δ



Toy example

• The minus signs for substitutions and indels assure that alignments

with many substitutions or indels will have low scores.

• We can then define the score S as the sum of individual scores at each

position:

S(WHAT/WH – Y) = 1 + 1 – δ – μ

• There is a more general way of describing the scoring process (not

necessary for "toy" problems such as the one above).

• In particular: write the target sequence (WHY) and the search space

(WHAT) as rows and columns of a matrix:



Toy example

• We have placed an x in the

matrix elements corresponding

to a particular alignment

• We have included one

additional row and one

additional column for initial

indels (-) to allow for the

possibility (not applicable here)

that alignments do not start at

the initial letters (W opposite W

in this case).



Toy example

• We can indicate the alignment

WHAT

WH-Y

as a path through elements of the matrix (arrows).

• If the sequences being compared were identical, then this path

would be along the diagonal.



Toy example

• Other alignments of WHAT with WHY would correspond to paths through

the matrix other than the one shown.

• Each step from one matrix element to another corresponds to the

incremental shift in position along one or both strings being aligned

with each other, and we could write down in each matrix element

the running score up to that point instead of inserting x (final score is

indicated in blue).



Toy example

• What we seek is the path through the matrix that produces the

greatest possible score in the element at the lower right-hand

corner.

• That is our "destination," and it corresponds to having used up all of

the letters in the search string (first column) and search space (first

row)-this is the meaning of global alignment.

• Using a scoring matrix such as this employs a particular trick of

thinking ...



Toy example

• For example, what is the "best" driving route from Los Angeles to St.

Louis? We could plan our trip starting in Los Angeles and then

proceed city to city considering different routes. For example, we

might go through Phoenix, Albuquerque, Amarillo, etc., or we could

take a more northerly route through Denver. We seek an itinerary

(best route) that minimizes the driving time. One way of analyzing

alternative routes is to consider the driving time to a city relatively

close to St. Louis and add to it the driving time from that city to St.

Louis.



Toy example

• Analyzing the best alignment using an alignment matrix proceeds

similarly, first filling in the matrix by working forward and then

working backward from the "destination" (last letters in a global

alignment) to the starting point.

• This general approach to problem-solving is called dynamic

programming.



Dynamic programming

• Dynamic programming; a computer algorithmic technique invented

in the 1940’s.

• Dynamic programming (DP) has applications to many types of

problems.

• Key properties of problems solvable with DP include that the

optimal solution typically contains optimal solutions to

subproblems, and only a “small” number of subproblems are

needed for the optimal solution.

(T.H. Cormen et al., Introduction to Algorithms, McGraw-Hill 1990).



Toy example (continued)

• We all agree that the best alignment is revealed by beginning at the

destination (lower right-hand corner matrix element) and working

backward, identifying the path that maximizes the score at the end.

• To do this, we will have to calculate scores for all possible paths.

into each matrix element ("city") from its neighboring elements

above, to the left, and diagonally above.

• We have now added row and column numbers to help us keep track

of matrix elements.



Toy example

• There are three possible paths into element (2, 2) (aligning left to

right with respect to both strings; letters not yet aligned are written

in parentheses):

Case a.

If we had aligned WH in WHY with W in WHAT (corresponding to element

(2, 1)), adding H in WHAT without aligning it to H in WHY corresponds to

an insertion of H (relative to WHY) and advances the alignment from

element (2, 1) to element (2,2) (horizontal arrow):

(W) H (AT)

(WH) - (Y)



Toy example

Case b.

If we had aligned W in WHY with WH in WHAT (corresponding to

element (1, 2)), adding the H in WHY without aligning it to H in

WHAT corresponds to insertion of H (relative to WHAT) and

advances the alignment from element (1, 2) to element (2, 2)

(vertical arrow):

(WH)-(AT)

(W)H (Y)



Toy example

Case c.

If we had aligned W in WHY with W in WHAT (corresponding to

element (1,1)), then we could advance to the next letter in both

strings, advancing the alignment from (1,1) to (2,2) (diagonal arrow

above):

(W)H(AT)

(W)H (y)

• Note that horizontal arrows correspond to adding indels to the string

written vertically and that vertical arrows correspond to adding indels

to the string written horizontally.



Toy example

• Associated with each matrix element (x, y) from which we could have

come into (2,2) is the score ��,� up to that point.

• Suppose that we assigned scores based on the following scoring

rules:

identity (match) +1

substitution (mismatch) -1

indel -2

• Then the scores for the three different routes into (2,2) are



Toy example

• The path of the cases a, b, or c that yields the highest score for ��,� is

the preferred one, telling us which of the alignment steps is best.

• Using this procedure, we will now go back to our original alignment

matrix and fill in all of the scores for all of the elements, keeping track

of the path into each element that yielded the maximum score to that

element.

• The initial row and column labelled by (-) corresponds to sliding

WHAT or WHY incrementally to the left of the other string without

aligning against any letter of the other string.

• Aligning (-) opposite (-) contributes nothing to the alignment of the

strings, so element (0,0) is assigned a score of zero.

• Since penalties for indels are -2, the successive elements to the right

or down from element (0, 0) each are incremented by -2 compared

with the previous one.



Toy example

• Thus ��,� is -2, corresponding to W opposite -, ��,� is -4,

corresponding to WH opposite -, etc., where the letters are coming

from WHAT.

• Similarly, ��,� is -2, corresponding to - opposite W, ��,� is -4,

corresponding to - opposite WH, etc., where the letters are coming

from WHY.

• The result up to this point is



Toy example

• Now we will calculate the score for (1,1). This is the greatest of ��,� +

1 (W matching W, starting from (0, 0)), ��,� - 2, or ��,� - 2.

• Clearly, ��,� + 1 = o + 1 = 1 "wins". (The other sums are -2 - 2 = -4.)

• We record the score value +1 in element (1, 1) and record an arrow that

indicates where the score came from.



Toy example

• The same procedure is used to calculate the score for element (2,1)

(see above).

• Going from (1,0) to (2,1) implies that H from WHY is to be aligned with

W of WHAT (after first having aligned W from WHY with (-)). This would

correspond to a substitution, which contributes -1 to the score. So

one possible value of ��,�= ��,�- 1 = -3.

• But (2, 1) could also be reached from (1, 1), which corresponds to

aligning H in WHY opposite an indel in WHAT (i.e., not advancing a

letter in WHAT). From that direction, ��,�= ��,�- 2 = 1 - 2 = -1.

• Finally, (2, 1) could be entered from (2,0), corresponding to aligning

W in WHAT with an indel coming after H in WHY. In that direction,

S�,�. = S�,� - 2 = -4 - 2 = -6.

• We record the maximum score into this cell (S2,1 = S1,1 - 2 = -1) and

the direction from which it came.



Toy example

• The remaining elements of the matrix are filled in by the same

procedure, with the following result:

• The final score for the alignment is �, = -1.

• The score could have been achieved by either of two paths (implied by

two arrows into (3, 4) yielding the same score).



Toy example

• The path through element (2,3) (upper path, bold arrows) corresponds

to the alignment

WHAT

WH-Y

which is read by tracing back through all of the elements visited in that

path.

• The lower path (through element (3, 3)) corresponds to the alignment

WHAT

WHY-

• Each of these alignments is equally good (two matches, one mismatch,

one indel).



Toy example

• Note that we always recorded the score for the best path into each

element.

• There are paths through the matrix corresponding to very "bad"

alignments. For example, the alignment corresponding to moving left

to right along the first row and then down the last column is

WHAT - - -

- - - -WHY

with score -14.

• For this simple problem, the computations were not tough. But when

the problems get bigger, there are so many different possible

aligmnents that an organized approach is essential.

• Biologically interesting alignment problems are far beyond what we

can handle with a No. 2.pencil and a sheet of paper, like we just did.



3 Global alignment

Formal development

• We are given two strings, not necessarily of the same length, but from

the same alphabet:

• Alignment of these strings corresponds to consecutively selecting each

letter or inserting an indel in the first string and matching that

particular letter or indel with a letter in the other string, or introducing

an indel in the second string to place opposite a letter in the first string.



Formal development

• Graphically, the process is represented by using a matrix as shown

below for n = 3 and m = 4:

• The alignment corresponding to the path indicated by the arrows is



Formal development

• Any alignment that can be written corresponds to a unique path

through the matrix.

• The quality of an alignment between A and B is measured by a score,

S(A, B), which is large when A and B have a high degree of similarity.

- If letters ai and bj are aligned opposite each other and are the

same, they are an instance of an identity.

- If they are different, they are said to be a mismatch.

• The score for aligning the first i letters of A with the first j letters of B is



Formal development

• S i,j is computed recursively as follows. There are three different ways that

the alignment of a1 a2 … ai with b1 b2 … bj can end:

where the inserted spaces "-" correspond to insertions or deletions

("indels") in A or B.

• Scores for each case are defined as follows:



Formal development

• With global alignment, indels will be added as needed to one or both

sequences such that the resulting sequences (with indels) have the

same length. The best alignment up to positions i and j corresponds to

the case a, b, or c before that produces the largest score for Si,j:

• The ''max'' indicates that the· one of the three expressions that yields

the maximum value will be employed to calculate S i,j



Formal development

• Except for the first row and first column in the alignment matrix, the

score at each matrix element is to ·be determined with the aid of the

scores in the elements immediately above, immediately to the left, or

diagonally above and to the left of that element.

• The scores for elements in the first row and column of the alignment

matrix are given by

• The score for the best global alignment of A with B is S(A, B) = Sn,m and

it corresponds to the highest-scoring path through the matrix and

ending at element ( n, m). It is determined by tracing back element by

element along the path that yielded the maximum score into each

matrix element.



Exercise

• What is the maximum score and corresponding alignment for aligning

A=ATCGT with B=TGGTG?

• For scoring, take

- �� , �� 1 if � � ��,

- �� , �� 1 if � � �� and

- �� , �� , �� 2



4 Local alignment

Rationale

• Proteins may be multifunctional. Pairs of proteins that share one of

these functions may have regions of similarity embedded in otherwise

dissimilar sequences.



Rationale

• With only partial sequence similarity and very different lengths,

attempts at global alignment of two sequences such as these would

lead to huge cumulative indel penalties.

• What we need is a method to produce the best local alignment; that

is, an alignment of segments contained within two strings (Smith and

Waterman, 1981).

• As before, we will need an alignment matrix, and we will seek a high-

scoring path through the matrix.

• Unlike before, the path will traverse only part of the matrix. Also, we

do not apply indel penalties if strings A and B fail to align “at the ends”.



Rationale

• Hence, instead of having elements -iδ and - jδ in the first row and first

column, respectively (-δ being the penalty for each indel), all the

elements in the first row and first column will now be zero.

• Moreover, since we are interested in paths that yield high scores over

stretches less than or equal to the length of the smallest string, there

is no need to continue paths whose scores become too small.

- If the best path to an element from its immediate neighbors above

and to the left (including the diagonal) leads to a negative score,

we will arbitrarily assign a 0 score to that element.

- We will identify the best local alignment by tracing back from the

matrix element having the highest score. This is usually not (but

occasionally may be) the element in the lower right-hand corner of

the matrix.



Mathematical formulation

• We are given two strings A = ala2a3 ... an and B = b1b2b3 … bm

• Within each string there are intervals I and J that have simillar sequences. I

and J are intervals of A and B, respectively [ I ⊂ A and J ⊂ B, where “⊂”

means “is an interval of”]

• The best local alignment score,M(A, B), for strings A and B is

where S(I, J) is the score for subsequences I and J and S(Ø,Ø) = 0.

• Elements of the alignment matrix are Mi,j, and since we are not

applying indel penalties at the ends of A and B, we write

��,� � ��,� � 0.




• The score up to and including the matrix element Mi,j is calculated by

using scores for the elements immediately above and to the left

(including the diagonal) ,but this time scores that fall below zero will be

replaced by zero.

• The scoring for matches, mismatches, and indels is otherwise the same

as for global alignment.

• The resulting expression for scoring Mi,j is




• The best local alignment is the one that ends in the matrix element

having the highest score:

• Thus, the best local alignment score for strings A and B is



Exercise

• Determine the best local alignment and the maximum alignment score for

A=ACCTAAGG and B=GGCTCAATCA

• For scoring, take

- �� , �� 2 if � � ��,

- �� , �� 1 if � � �� and

- �� , �� , �� 2

• The best local alignment is given below. The “aligned regions” of A and B

are indicated by the blue box

A: ACCT – AAGG -

B: GGCTC AATC A



Scoring rules

• We have used alignments of nucleic acids as illustrative examples, but

it should be noted that for protein alignments the scoring is much

more complicated.

• Hence, at this point, we address briefly the issue of assigning

appropriate values to s(ai,bj), s(ai,-), and s(-,bj) for nucleotides.

• For scoring issues in case of amino acids, you can find more details in

Deonier et al. 2005 (Chapter 7, p182-189).



Scoring rules

• Considering s(ai, bj) first, we write down a scoring matrix containing

all possible ways of matching ai with bj, ai, bj ∈ {A, C, G, T} and write in

each element the scores that we have used for matches +1 and

mismatches -1.

• Note that this scoring matrix contains the assumption that aligning A

with G is just as bad as aligning A with T because the mismatch

penalties are the same in both cases.



Scoring rules

• A first issue arises when observing that studies of mutations in

homologous genes have indicated that

transition mutations (A → G, G → A, C → T, or T → C) occur

approximately twice as frequently as do

transversions (A → T, T → A, A → C, G → T, etc.). (A, G are purines,

larger molecules than pyrimidines T, C)

• Therefore, it may make sense to apply a lesser penalty for transitions

than for transversions

• The collection of s(ai , bj) values in that case might be represented as



Scoring rules

• A second issue relates to the scoring of gaps (a succession of indels),

in that sense that we never bothered about whether or not indels are

actually independent.

• Up to now, we have scored a gap of length k as

ω(k) = -kδ

• However, insertions and deletions sometimes appear in "chunks" as a

result of biochemical processes such as replication slippage.

• Also, deletions of one or two nucleotides in protein-coding regions

would produce frameshift mutations (usually non-functional – see

before), but natural selection might allow small deletions that are

integral multiples of 3, which would preserve the reading frame and

some degree of function.



Scoring rules

• The aforementioned examples suggest that it would be better to

have gap penalties that are not simply multiples of the number of

indels.

• One approach is to use an expression such as

ω(k) = -α-β(k – 1)

• This would allow us to impose a larger penalty for opening a gap (-α)

and a smaller penalty for gap extension (-β for each additional base in

the gap).



5 Rapid alignment methods

5.a Introduction

• The need for automatic (and rapid!) approaches also arises from the

interest in comparing multiple sequences at once (and not just 2)

• Recall that A = a1a2 ... ai and B = b1b2 ... bj have alignments that can

end in three possibilities: (-,bj), (ai,bj), or (ai, -).

• The number of alternatives for aligning ai with bj can be calculated as

3 = 22 – 1

• If we now introduce a third sequence c1c2…ck, the three-sequence

alignment can end in one of seven ways (7 = 23 -1): (ai,-,-),(-,bj,-),

(-,-,ck),(-,bj,ck),(ai,-,ck),(ai,bj,-), and (ai, bj,ck) In other words, the

alignment ends in 0, 1, or 2 indels.



5.b Search space reduction

• We can reduce the search space by analyzing word content (see Chapter 3).

Suppose that we have the query string I indicated below:

• This can be broken down into its constituent set of overlapping k-tuples.

For k = 8, this set is



Search space reduction

• If a string is of length n, then there are n - k + 1 k-tuples that are produced

from the string. If we are comparing string I to another string J (similarly

broken down into words), the absence of anyone of these words is

sufficient to indicate that the strings are not identical.

- If I and J do not have at least some words in common, then we can

decide that the strings are not similar.

• We know that when P(A) = P(C) = P(G) = P(T) = 0.25, the probability that an

octamer beginning at any position in string J will correspond to a particular

octamer in the list above is 1/48.

- Provided that J is short, this is not very probable.

- If J is long, then it is quite likely that one of the eight-letter words in I

can be found in J by chance.

• The appearance of a subset of these words is a necessary but not sufficient

condition for declaring that I and J have meaningful sequence similarity.



Word Lists and Comparison by Content

• Rather than scanning each sequence for each k-word, there is a way to

collect the k-word information in a set of lists.

• A list will be a row of a table, where the table has 4k rows, each of which

corresponds to one k-word.

• For example, with k = 2 and the sequences below,

we obtain the word lists shown in the table on the next slide (Table 7.2,

Deonier et al 2005).




• Thinking of the rows as k-words, we denote the list of positions in the row

corresponding to the word w as Lw(J)

- e.g., with w = CG, LCG(J) = {5,11}

• These tables are sparse, since the sequences are short

• Time is money: limit detailed comparisons only to those sequences that

share enough "content"

Content sharing = = k-letter words in common




• The statistic that counts k-words in common is

where Xi,j = 1 if IiIi+1...Ii+k-1 = JjJj+1 … Jj+k-1 and 0 otherwise.

• The computation time is proportional to n x m, the product of the sequence

lengths.

• To improve this, note that for each w in I, there are #Lw(J) occurrences in J.

So the sum above is equal to:

• Note that this equality is a restatement of the relationship between + and x




• The second computation is much easier.

- First we find the frequency of k-letter words in each sequence. This is

accomplished by scanning each sequence (of lengths n and m).

- Then the word frequencies are multiplied and added.

• For our sequence of numbers of 2-word matches, the statistic above is

• If 10 is above a threshold that we specify, then a full sequence comparison

can be performed.

- Low thresholds require more comparisons than high thresholds.




• This aforementioned method is quite fast, but the comparison totally

ignores the relative positions of the k-words in the sequence.

• A more sensitive method is needed....

• Also, what if you have a word list of over 5000 entries and you are looking

at k’s of more than 10? Is there a more efficient way of browsing through

the list?

• Suppose I = GGAATAGCT, J = GTACTGCTAGCCAAATGGACAATAGCTACA, and

we wish to find all k-word matches between the sequences with k = 4.

• Our method using k = 4 depends on putting the 4-words in J into a list

ordered alphabetically as in the table below (Table 7.1.; Deonier et al 2005



5.c Binary Searches

• Beginning with GGAA, we look in the J list for the 4-words contained in I by

binary search.

• Since list J (of length m = 25) is stored in a computer, we can extract the

entry number m/2, which in this example is entry 13, CTAC.

• In all cases we round fractions up to the next integer.

• Then we proceed as follows:



Binary Searches

• Step 1: Would GGAA be found before entry 13 in the alphabetically sorted

list?

- Since it would not, we don’t need to look at the first half of the list.

• Step 2: In the second half of the list, would GGAA occur before the entry at

position m/2 + m/4

- i.e., before entry (18.75 hence) 19, GGAC

- GGAA would occur before this entry, so that after only two

comparisons -we have eliminated the need to search 75% of the list and

narrowed the search to one quarter of the list.

• Step 3: Would GGAA occur after entry 13 but at or before entry 16?

- We have split the third quarter of the list into two m/8 segments.

- Since it would appear after entry 16 but at or before entry 19, we need

only examine the three remaining entries.

• Steps 4 and 5: Two more similar steps are needed to conclude that GGAA is

not contained in J.



Binary Searches

• Had we gone through the whole ordered list sequentially, 19 steps would

have been required to determine that the word GGAA is absent from J.

• With the binary search, we used only five steps.

• We proceed in similar fashion with the next 4-word from I, GAAT. We also

fail to find this word in J, but the word at position 3 of I, AATA, is found in

the list of 4-words from J, corresponding to position 21 of J.

• Words in I are taken in succession until we reach the end.

• Remark:

With this method, multiple matchings are found. This process is analogous

to finding a word in a dictionary by successively splitting the remaining

pages in half until we find the page containing our word.



Binary Searches

• In general, if we are searching a list of length m starting at the top and

going item by item, on average we will need to search half the list before

we find the matching word (if it is present).

• If we perform a binary search as above, we will need only log2(m) steps in

our search.

• This is because m = 2Iog2(m)

, and we can think of all positions in our list of

length m as having been generated by log2(m) doublings of an initial

position.

• In the example above, m=30. Since 32 = 25, we should find any entry after

five binary steps. Note log2(30)=4.9



Toy example

• Suppose a dictionary has 900 pages

• Then finding the page containing the 9-letter word “crescendo” in this

dictionary, using the binary search strategy, should require how many

steps?

- The dictionary is 864 pages long say

- 29= 512

- 210

= 1024

- 512 < 864 < 1024; log2(864) = 9.75 ~10

• You need 10 binary steps to find the appropriate page...

• Within the page, you can again adopt a binary search to find the correct

word among the list of words on that page ...

• Although binary searches are very efficient, we could use some automated

steps to reduce the search space.



5.d Looking for regions of similarity using FASTA

• FASTA (Pearson and Lipman, 1988) is a rapid alignment approach that com-

bines methods to reduce the search space

• FASTA (pronounced FAST-AYE) stands for FAST-ALL, reflecting the fact that

it can be used for a fast protein comparison or a fast nucleotide

comparison.

• It depends on k-tuples and (Smith-Waterman) local sequence alignment.

• As an introduction to the rationale of the FASTA method, we begin by

describing dot matrix plots, which are a very basic and simple way of

visualizing regions of sequence similarity between two different strings. It

allows us to identify k-tuple correspondences (first crucial step in FASTA)



Dot Matrix Comparisons

• Dot matrix comparisons are a special type of alignment matrix with

positions i in sequence I corresponding to rows, positions j in sequence J

corresponding to columns

• Moreover, sequence identities are indicated by placing a dot at matrix

element (i,j) if the word or letter at Ii is identical to the word or letter at Jj.

• An example for two DNA strings is

shown in the right panel. The

string CATCG in I appears twice in

J, and these regions of local

sequence similarity appear as two

diagonal arrangements of dots:

diagonals represent regions having

sequence similarity.



Dot Matrix Comparisons

• When I and J are DNA sequences and are short, the patterns of this type are

relatively easy to see.

• When I and J are DNA sequences and very long, there will be many dots in

the matrix since, for any letter at position j in J, the probability of having a

dot at any position i in I will equal the frequency of the letter Jj in the DNA.

• When I and J are proteins, dots in the matrix elements record matches

between amino acid residues at each particular pair of positions.



FASTA: Rationale

• How is this used within FASTA

(Wilbur and Lipman, 1983)?

• The rationale for FASTA can be

visualized by considering what

happens to a dot matrix plot when

we record matches of k-tuples (k >

1)

instead of recording matches of

single letters

(Fig. 7.1; Deonier et al 2005).



FASTA: Rationale

• We again place entries in the alignment matrix, except this time we only

make entries at the first position of each dinucleotide or trinucleotide (k-

tuple matches having k = 2 (plotted numerals 2) or k = 3 (plotted numerals

3).

• By looking for words with k > 1, we find that we can ignore most of the

alignment matrix since the absence of shared words means that sub-

sequences don't match well.

- There is no need to examine areas of the alignment matrix where there

are no word matches. Instead, we only need to focus on the areas

around any diagonals.

• Our task is now to compute efficiently diagonal sums of scores, Sl, for

diagonals and the question is how can we compute these scores?



The number of matrix entries is reduced as k increases.



FASTA: Rationale

• Consider again the two strings I

and J that we used before:

• Scores can be computed in the

following way:

- Make a k-word list for J.



FASTA: Rationale

- Then initialize all row sums to 0:

(Why does l not range from -11 to 7?)

- Next proceed with the 2-words of I, beginning with i = 1, GC. Looking in

the list for J, we see that LGC(J)={6}, so we know that at l = 1 -6 = -5 there

is a 2-word match of GC.

� Therefore, we replace S-5= 0 by S-5 = 0 + 1 = 1.

- Next, for i = 2, we have LCA(J)={2,8}.

� Therefore replace S2-2 = S0 = 0 by S0 = 0 + 1, and replace S2-8 = S-6 = 0

by S-6 = 0 + 1.



FASTA: Rationale

- These operations, and the operations for all of the rest of the 2-words

in I, are summarized below.

� Note that for each successive step, the then-current score at Si is

employed: S0 was set to 1 in step 2, so l is incremented by 1 in step

3.



FASTA: Rationale



FASTA: Rationale

• In conclusion, in our example there are 7 x 11 = 77 potential 2-matches, but

in reality there are ten 2-matches with four nonzero diagonal sums.

Where do 7 and 11 come from?

• We have indexed diagonals by the offset, l = i - j.

• In this notation, the nonzero diagonal sums are S+1 = 1, S0 = 4,

S-5 = 1, and S-6 = 4.

• Notice that we only performed additions when there were 2-word matches.

• It is possible to find local alignments using a gap length penalty of -gx for a

gap of length x along a diagonal.

• We are now in good shape to understand the 5 steps involved in FASTA

aligning



FASTA steps

Five steps are involved in FASTA:

• Step 1: Use the look-up table to identify k-tuple identities between I and J.

• Step 2: Score diagonals containing k-tuple matches, and identify the ten

best diagonals in the search space.

• Step 3: Rescore these diagonals using an appropriate scoring matrix

(especially critical for proteins), and identify the subregions with the highest

score (initial regions).

• Step 4: Join the initial regions with the aid of appropriate joining or gap

penalties for short alignments on offset diagonals.

• Step 5: Perform dynamic programming alignment within a band

surrounding the resulting alignment from step 4.



FASTA steps in more detail

• Step 1: identify k-type identities

- To implement the first step, we pass through I once and create a table

of the positions i for each possible word of predetermined size k.

- Then we pass through the search space J once, and for each k-tuple

starting at successive positions j, "look up" in the table the

corresponding positions for that k-tuple in I.

- Record the i,j pairs for which matches are found: the i,j pairs define

where potential diagonals in the alignment matrix can be found.




• Step 2: identify high-scoring diagonals

- If I has n letters and J has m letters, then there are n+m-1 diagonals.

� Think of starting in the upper left-hand corner, drawing successive

diagonals all the way down, moving your way through the matrix

from left to right (m diagonals).

� Start drawing diagonals through all positions in I (n diagonals).

� Since you will have counted the diagonal starting at (1,1) twice, you

need to subtract 1.

- Note that we have seen before how to score diagonals (in particular, in

our examples, without accounting for gaps).




• Step 2: identify high-scoring diagonals (continued)

- To score the diagonals, calculate the number of k-tuple matches for

every diagonal having at least one k-tuple (identified in step 1).

� Scoring may take into account distances between matching k-tuples

along the diagonal.

� Note that the number of diagonals that needs to be scored will be

much less than the number of all possible diagonals (reduction of

search space).

- Identify the significant diagonals as those having significantly more k-

tuple matches than the mean number of k-tuple matches.

� For example, if the mean number of 6-tuples is 5 ± 1, then with a

threshold of two standard deviations, you might consider diagonals

having seven or more 6-tuple matches as significant.

- Take the top ten significant diagonals.




• Step 3: Rescore these diagonals

- We rescore the diagonals using a scoring table to find subregions with

identities shorter than k.

- Rescoring reveals sequence similarity not detected because of the

arbitrary demand for uninterrupted identities of length k.

- The need for this rescoring is illustrated by the two examples below.

� In the first case, the placement of mismatches spaced by three

letters means that there are no 4-tuple matches, even though the

sequences are 75% identical.

� The second pair shows one 4-tuple match, but the two sequences

are only 33% identical.

- We retain the subregions with the highest scores.




• Step 4: Joining diagonals



Offset diagonals

• Diagonals may be offset from each other, if there were a gap in the

alignment (i.e., vertical or horizontal displacements in the alignment matrix,

as described in the previous chapter).

• Such offsets my indicate indels, suggesting that the local alignments

represented by the two diagonals should be joined to form a longer

alignment.

• Diagonal dl is the one having k-tuple matches at positions i in string I and j

in string J such that i – j = l. As described before, l = i – j is called the offset.



Offset diagonals

• The idea is to find the best-scoring combination of diagonals

• Two offset diagonals can be joined with a gap, if the resulting alignment has

a higher score

• Note that different gap open and extension penalties may be are used




• Step 5: Smith- Waterman local alignment

- The last step of FASTA is to perform local alignment using dynamic

programming round the highest-scoring

- The region to be aligned covers –w and +w offset diagonal to the

highest scoring diagonals

- With long sequences, this region is typically very small compared to the

entire nxm matrix (hence, once again, a reduction of the search space

was obtained)




In FASTA, the alignment step can be restricted to a comparatively narrow

window extending +w to the right and -w to the left of the positions included

within the highest-scoring diagonal (dynamic programming)



The Smith-Waterman algorithm (local alignment)

• Needleman and Wunsch (1970) were the first to introduce a heuristic

alignment algorithm for calculating homology between sequences (global

alignment).

• Later, a number of variations have been suggested, among others Sellers

(1974) getting closer to fulfilling the requests of biology by measuring the

metric distance between sequences [Smith and Waterman, 1981].

• Further development of this led to the Smith-Waterman algorithm based

on calculation of local alignments instead of global alignments of the

sequences and allowing a consideration of deletions and insertions of

arbitrary length.



The Smith-Waterman algorithm

• The Smith-Waterman algorithm uses individual pair-wise comparisons

between characters as:

Do you recognize this formula?

• The Smith-Waterman algorithm is the most accurate algorithm when it

comes to search databases for sequence homology but it is also the most

time consuming, thus there has been a lot of development and suggestions

for optimizations and less time-consuming models. One example is BLAST.



Properties of FASTA

• Fast compared to local alignment only using dynamic programming as such

- A narrow region of the full alignment matrix is aligned

- With long sequences, this region is typically very small compared to the

whole matrix

• Increasing the parameter k (word length), decreases the number of hits

- It increases specificity

- It decreases sensitivity

• FASTA can be very specific when identifying long regions of low similarity

- Specific method does not produce many incorrect results

- Sensitive method produces many of the correct results

More info at http://www.ebi.ac.uk/fasta

(parameter ktup in the software corresponds to the parameter k in the class

notes)



Sensitivity and specificity reminder

• These concepts come from the world of “clinical test assessment”

- TP = true positives; FP = false positives

- TN = true negatives; FN = false negatives

• Sensitivity = TP/(TP+FN)

• Specificity = TN/(TN+FP)

Signal present Signal not present

Signal detected TP FP

Signal not detected FN TN



FASTA in practice

• Help on nucleotide searches:

http://www.ebi.ac.uk/2can/tutorials/nucleotide/fasta.html

• Help on interpretation of such search results:

http://www.ebi.ac.uk/2can/tutorials/nucleotide/fasta1.html



5.e BLAST

• The most used database search programs are BLAST and its descendants.

• BLAST is modestly named for Basic Local Alignment Search Tool, and it was

introduced in 1990 (Altschul et al., 1990).

• Whereas FASTA speeds up the search by filtering the k-word matches,

BLAST employs a quite different strategy (see later).



BLAST steps

In particular, three steps are involved in BLAST:

• Step 1: Find local alignments between the query sequence and a data base

sequence (“seed hits”)

• Step 2: Extend the seed hits into high-scoring local alignments

• Step 3: Calculate p-values and a rank ordering of the local alignments



BLAST steps in more detail

• Step 1: Finding local matches

- First, the query sequence is used as a template to construct a set of

subsequences of length k that can score at least T when compared with

the query.

- Take as before

- To construct a “neighbourhood” of GCATC (collection of words of

length 5), we specify for instance T=4: (scoring matches 1, mismatches

0)



• Each of these terms (“seeds”) represents three sequences that at least

score T=4 when compared to GCATC.

• So in total there are 1 + (3 x 5) = 16 exact matches to search for in J.

• For the three other 5-word patterns in I (CATCG, ATCGG, and TCGGC), there

are also 16 exact 5-words, for a total of 4 x 16 = 64 5-word patterns to

locate in J.

• A hit is defined as an instance in the search space (database) of a k-word

match, within threshold T, of a k-word in the query sequence.

In actual practice, the hits correspond to a tiny fraction of the entire search

space.



• Step 2: Extending the alignment, starting from the” seed” hits

- Starting from any seed hit, the extension includes successive positions,

with corresponding increments to the alignment score.

- Unlike the earlier version of BLAST, gaps can be accommodated in the

later versions.

� With the original version of BLAST, over 90% of the computation

time was employed in producing the un-gapped extensions from

the hits.

� This is because the initial step of identifying the seed hits was

effective in making this alignment tool very fast.

� Later versions of BLAST require the same amount of time to find

the seed hits and have reduced the time required for the un-

gapped extensions considerably.

- The newer versions of BLAST run about three times faster than the

original version (Altschul et aL, 1997).



• Step 3: Assess significance

- Another aspect of a BLAST analysis is to rank-order by p-values the se-

quences found

- If the database is D and a sequence X scores S(D,X) = s against the

database, the p-value is P(S(D,Y)≥ s), where Y is a random sequence.

- The smaller the p-value, the greater the "surprise" and hence the

greater the belief that something real has been discovered.

- A p-value of 0.1 means that with a collection of query sequences picked

at random, in 1/10 of the instances a score that is as large or larger

would be discovered.

- A p-value of 10-6

means that only once in a million instances would a

score of that size appear by chance alone.



E-values and p-values

• P-value; probability value:

this is the probability that a hit would attain at least the given score, by

random chance for the search database

• E value; expectation value:

this is the expected number of hits of at least the given score that you

would expect by random chance for the search database

There is a theoretical foundation that shows that, when the E-value is small

enough, it is essentially a p-value (small enough ~E<0.10)



E-values and p-values

• E-value < 10e-100: Identical sequences.

- You will get long alignments across the entire query and hit sequence.

• 10e-50 < E-value < 10e-100: Almost identical sequences.

- A long stretch of the query protein is matched to the database.

• 10e-10 < E-value < 10e-50: Closely related sequences.

- Could be a domain match or similar.

• 1 < E-value < 10e-6: Could be a true homologue but it is a gray area.



BLAST properties

• BLAST is extremely fast. It can be on the order of 50-100 times faster than

the Smith-Waterman (local alignment) approach

• Its main idea is built on the conjecture that homologous sequences are

likely to contain a short high-scoring similarity region, a hit. Each hit gives a

seed that BLAST tries to extend on both sides.

• It is preferred over FASTA for large database searches

• There exists a statistical theory for assessing significance



BLAST properties

• Many variants exist to the initial BLAST theme ... Depending on the nature

of the sequence it is possible to use different BLAST programs for the

database search.

• There are five versions of the BLAST program: BLASTN, BLASTP, BLASTX,

TBLASTN,TBLASTX:



BLAST in practice

• Relevant questions include:

- What inferences about this sequence can you make from this

information?

- What is the identity of the sequence?

- What gene do you think it encodes?

- What organism do you think it comes from?

- How reliable do you think this inference is? Why?

• More info via

http://www.ncbi.nlm.nih.gov/BLAST/blast_overview.html



Should I use Smith-Waterman or other algorithms for sequence similarity

searching?

• The Smith-Waterman algorithm is quite time demanding because of the

search for optimal local alignments, and it also imposes some requirements

on the computer's memory resources as the comparison takes place on a

character-to-character basis.

• The fact that similarity searches using the Smith-Waterman algorithm take

a lot of time often prevents this from being the first choice, even though it

is the most precise algorithm for identifying homologous regions between

sequences.

• A combination of the Smith-Waterman algorithm with a reduction of the

search space (such as k-word identification in FASTA) may speed up the

process.




searching?

• BLAST and FASTA are heuristic approximations of dynamic programming

algorithms. These approximations are less sensitive (then f.i. Smith-

Waterman) and do not guarantee to find the best alignment between two

sequences. However, these methods are not as time-consuming as they

reduce computation time and CPU usage [Shpaer et al., 1996].

• Hence, the researcher needs to make a choice between

- Having a fast and effective data analysis (BLAST)

- Reducing the risk of missing important information by using the most

sensitive algorithms for data base searching (Smith-Waterman / FASTA)




searching?

• Through the Japanese Institute of Bioinformatics Research and

Development (BIRD) a public available software version of Smith-

Waterman, SSEARCH, is accessible: http://www-btls.jst.go.jp/cgi-

bin/Tools/SSEARCH/index.cgi.

• There are also commercial software packages available which perform

Smith-Waterman searches.

• Remember that the result of a Smith-Waterman algorithm searching will be

only returning one result for each pair of compared sequences: the most

optimal alignment



References:

• Deonier et al. Computational Genome Analysis, 2005, Springer.

(Chapters 6,7)

Preparatory Reading:

• Foster et al 2004. Beyond race : towards a whole genome perspective on human populations

and genetic variation. Nature Reviews Genetics 5: 790-.

• Balding 2006. A tutorial on statistical methods for population association studies. Nature

Reviews Genetics 7: 781-.

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